New features on real zeros of random polynomials

Abstract Let Q n ( x ) = ∑ k = 0 n A k x k be a random algebraic polynomial in which the coefficients A 0 , A 1 , A 2 , … form a sequence of independent normally distributed random variables. In this work we study the behavior of the expected density of real zeros of Q n ( x ) for the case that the variances of the middle coefficients are substantially large, say Var ( A k ) = ρ ( k − n / 2 ) 2 . We find some new and interesting features about the distribution and the expected number of real zeros of such a polynomial for different values of ρ . We also consider the case where the variances of the coefficients are decreasing as Var ( A k ) = e − k 2 / 2 n 7 / 4 , and we show that the asymptotic behavior of the expected number of real zeros of Q n ( x ) is of order n 3 / 8 .